We construct a non-commutative version of the Grassmann variety G(2,4) as a non-commutative moduli space of linear subspaces in a projective space.

We formulate an effective variant of the Yau–Tian–Donaldson conjecture and then review effective results on K-stability of spherical varieties, that is, K-stability criteria which can be effectively computed given the combinatorial data associated with the variety. We focus on the standard notion of K-stability as defined by Donaldson for constant scalar curvature Kähler metrics.

This is a survey on what is known, up to date, on normal subgroups of Cremona groups. There are several different approaches to showing that they exist, and we will take a look at each of them, more or less in chronological order.

For a locally free sheaf \mathcal{E} on a smooth projective curve, we can define the punctual Quot scheme which parametrizes torsion quotients of \mathcal{E} of length n supported at a fixed point. It is known that the punctual Quot scheme is a normal projective variety with canonical Gorenstein singularities. In this note, we show that the punctual Quot scheme is a \mathbb{Q} -factorial Fano variety of Picard number one.

In this article, we give an introduction to asymptotic stability in two dimensional incompressible flows, and a non-technical overview of the recent proof of uniform-in-viscosity inviscid damping and vorticity depletion near periodic shear flows on a non-square torus.

A conjecture of Boone and Higman from the 1970’s asserts that a finitely generated group G has solvable word problem if and only if G can be embedded into a finitely presented simple group. We comment on the history of this conjecture and survey recent results that establish the conjecture for many large classes of interesting groups.

In this note, we report some recent progress on the Jordan property for (birational) automorphism groups of projective varieties and compact complex varieties.

In this paper, we rewrite the time-dependent Bogoliubov–de Gennes (BdG) equation in an appropriate semiclassical form and establish its semiclassical limit to a two-particle kinetic transport equation with an effective mean-field background potential satisfying the one-particle Vlasov equation. Moreover, for some semiclassical regimes, we obtain a higher-order correction to the two-particle kinetic transport equation, capturing a nontrivial two-body interaction effect. The convergence is proven for C^{2} interaction potentials in terms of a semiclassical optimal transport pseudo-metric.Furthermore, combining our current results with the results of Marcantoni et al. [Ann. Henri Poincaré (2024)], we establish a joint semiclassical and mean-field approximation of the dynamics of a system of spin- \frac{1}{2} Fermions by the Vlasov equation in some weak topology.

We consider a modulated magnetic field, B(t) = B_{0} + \varepsilon f(\omega t) , perpendicular to a fixed plane, where B_{0} is constant, \varepsilon>0 and f a periodic function on the torus {\mathbb{T}}^{n} . Our aim is to study classical and quantum dynamics for the corresponding Landau Hamiltonian. It turns out that the results depend strongly on the chosen gauge. For the Landau gauge the position observable is unbounded for “almost all” non-resonant frequencies \omega . On the contrary, for the symmetric gauge we obtain that, for “almost all” non-resonant frequencies \omega , the Landau Hamiltonian is reducible to a two-dimensional harmonic oscillator and thus gives rise to bounded dynamics. The proofs use KAM algorithms for the classical dynamics. Quantum applications are given. In particular, the Floquet spectrum is absolutely continuous in the Landau gauge while it is discrete, of finite multiplicity, in symmetric gauge.

The aim of this paper is to survey and complete, mostly by numerical simulations, results on a remarkable Boussinesq system describing weakly nonlinear, long surface water waves. It is the only member of the so-called ( abcd ) family of Boussinesq systems known to be completely integrable.